C4 parametric curves_lesson

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A2-level maths UK

C4 parametric curves_lesson

  1. 1. A2 Mathematics: C4 Core Maths<br /> Curves and Tangents<br />Parametric Curves<br />
  2. 2. Objectives<br />We will be able to <br />Plot Graphs defined by parametric equations<br />by hand and <br />by calculator<br />Use algebra to eliminate the parameter and find the Cartesian equation of the curve.<br />Find the gradient of the curve for any value of the parameter.<br />Find the equation of the tangent or normal to the curve at any value of the parameter.<br />
  3. 3. What is a Parametric Graph ? <br />To plot a graph<br />we could follow a point <br />as it crawls <br />along the curve<br />especially<br />If the point obeys a rule<br />If it gives x and y<br />In terms of time <br />Or other parameter<br />
  4. 4. Tracing out a Parametric Graph <br />http://www.flashandmath.com/mathlets/calc/param2d/param_advanced.html<br />
  5. 5. Tracing out a Parametric Graph <br />This also shown in your WEC text-book<br />On page 323<br />
  6. 6. Parametric Curve examples<br />
  7. 7. Parametric Curve examples<br />
  8. 8. Parametric Curve examples<br />
  9. 9. Parametric Equations for a Curve<br />x=2t, y=15t– 5t²<br />
  10. 10. Plotting x and y via parameters<br />
  11. 11. Curves defined by parametric equations<br />
  12. 12. Parametric Equations for a Curve<br />x = 3cosθ, y = 3sinθ<br />
  13. 13. Plotting x and y via parameters<br />
  14. 14. Curves defined by parametric equations<br />
  15. 15. Plotting Parametric Curves<br />Use Sharp EL9900 calculator<br />Parametric settings on next slide<br />Use Autograph<br />Equation entry via x=2t, y=t^2<br />Separated by a comma<br />
  16. 16. Parametric Settings for EL9900<br />
  17. 17. Parametric Entry for EL9900<br />
  18. 18. Parametric Displays on the EL9900<br />
  19. 19. Cartesian Equation for a Curve<br />We have <br />x as a function of t or θ<br /> And<br />yas a function of t or θ<br />We need to eliminate t or θ<br />Leaving only x and y.<br />Methods<br />Eliminate t by substitution and algebra<br />Eliminate θ via trigonometric Identities and algebra<br />
  20. 20. Cartesian Equation – Eliminate t<br />x = t2, y = t – t2<br />
  21. 21. Cartesian Equation – identities in θ<br />x = 3cosθ, y = sinθ<br />
  22. 22. Activity step 1<br />Use table of values to plot each curve (and/or use your calculator).<br />Match each parameter formula and its curve with correct curve card.<br />Match each curves with its correct Cartesian equation.<br />
  23. 23. Parametric Equations for a Curve<br />x = 3cosθ, y = 3sinθ<br />
  24. 24. Cartesian Equation for a Curve<br />x2 + y2 = 9<br />
  25. 25. Curves defined by parametric equations<br />
  26. 26. Parametric Equations for a Curve<br />x=2t, y=15t– 5t²<br />
  27. 27. Cartesian Equation for a Curve<br />4y = 15x– 4.9x2<br />
  28. 28. Curves defined by parametric equations<br />
  29. 29. Parametric Equations for a Curve<br /> x=t²–4, y=t³–4t<br />
  30. 30. Cartesian Equation for a Curve<br />y = x√(x+4)<br /> y = x(x+4)0.5<br />
  31. 31. Curves defined by parametric equations<br />
  32. 32. Parametric Equations for a Curve<br />x=sinθ, y=sin2θ<br />
  33. 33. Cartesian Equation for a Curve<br />y = 2x√(1-x2)<br />y = 2x(1-x2)0.5<br />
  34. 34. Curves defined by parametric equations<br />
  35. 35. Parametric Equations for a Curve<br />x=t2, y=t3<br />
  36. 36. Cartesian Equation for a Curve<br />y=x√x<br />
  37. 37. Curves defined by parametric equations<br />
  38. 38. Parametric Equations for a Curve<br />x=t, y=1/t<br />
  39. 39. Cartesian Equation for a Curve<br /> y = 1/x<br />
  40. 40. Curves defined by parametric equations<br />
  41. 41. Parametric Equations for a Curve<br />x = 1+ t, y = 2 - t<br />
  42. 42. Cartesian Equation for a Curve<br />x + y = 3<br />
  43. 43. Curves defined by parametric equations<br />
  44. 44. Parametric Equations for a Curve<br />x=(2+3t)/(1+t), y=(3–2t)/(1+t)<br />
  45. 45. Parametric Equations for a Curve<br /> y=13–5x<br />
  46. 46. Curves defined by parametric equations<br />Stops here !<br />
  47. 47. Extension: Try these Parameters<br />x= t + 1/t, y= t - 1/t<br />x = 3cosθ, y= sinθ<br />Investigate/Create your own<br />
  48. 48. Parametric Equations for a Curve<br />x=………...., y=…….……..<br />
  49. 49. Curves defined by parametric equations<br />
  50. 50. Tangents to the curve?<br />How do we find dy/dx ?<br />How do we find the equation of the tangent at one particular point on the curve <br />– for example when t=1<br />
  51. 51. Parametric Equations for a Curve<br />x = 3cosθ, y = 3sinθ<br />
  52. 52. Gradient of Tangents to the Curve<br />We know (why?) that<br />
  53. 53. Gradient of Tangents to the Curve<br />x = 3cosθ, y = 3sinθ<br />...so.....<br />
  54. 54. Gradient of Tangents to the Curve<br />Putting it together.......<br />
  55. 55. How do we find a particular tangent?<br />Given a particular t value <br /> find x and y, and dy/dx<br />Now we have <br /><ul><li>the gradient of the tangent and
  56. 56. the co-ordinates where it touches the curve</li></ul>.......so.....<br />
  57. 57. Equation of one Tangent to Circle<br />
  58. 58. Equation of one Tangent to Circle<br /> x = 3cosπ/4, y = 3sin π/4<br /> ...so.....<br />
  59. 59. Image of one Tangent to the Curve<br />
  60. 60. Activity step 2<br />Use x and y parameter functions, to match <br />dy/dx equation<br /> one tangent equation <br />with previous cards <br />
  61. 61. Parametric Equations for a Curve<br />x=2t, y=15t– 5t²<br />
  62. 62. Gradient of Tangents to the Curve<br />
  63. 63. Image of one Tangent to the Curve<br />
  64. 64. Equation of one Tangent to the Curve<br />
  65. 65. Parametric Equations for a Curve<br /> x=t²–4, y=t³–4t<br />
  66. 66. Gradient of Tangents to the Curve<br />
  67. 67. Image of one Tangent to the Curve<br />
  68. 68. Equation of One Tangent to the Curve<br />
  69. 69. Parametric Equations for a Curve<br />x=sinθ, y=sin2θ<br />
  70. 70. Gradient of Tangents to the Curve<br />
  71. 71. Image of Tangent to Curve<br />
  72. 72. Equation of One Tangent to the Curve<br />
  73. 73. Parametric Equations for a Curve<br />x=t2, y=t3<br />
  74. 74. Gradient of Tangents to the Curve<br />
  75. 75. Image of Tangent to Curve<br />
  76. 76. Equation of one Tangent to the Curve<br />
  77. 77. Parametric Equations for a Curve<br />x=t, y=1/t<br />
  78. 78. Gradient of Tangents to the Curve<br />
  79. 79. Image of Tangent to Curve<br />
  80. 80. Equation of one Tangent to the Curve<br />
  81. 81. Parametric Equations for a Curve<br />x = 1+ t, y = 2 - t<br />
  82. 82. Gradient of Tangents to the Curve<br />
  83. 83. Image of Tangent to Curve<br />
  84. 84. Equation of one Tangent to the Curve<br />
  85. 85. Parametric Equations for a Curve<br />x=(2+3t)/(1+t), y=(3–2t)/(1+t)<br />
  86. 86. Gradient of Tangents to the Curve<br />
  87. 87. Image of Tangent to Curve<br />
  88. 88. Equation of one Tangent to the Curve<br />
  89. 89. Parametric Equations for a Curve<br />x=………...., y=…….……..<br />
  90. 90. Curves defined by parametric equations<br />
  91. 91. Gradient of Tangents to the Curve<br />
  92. 92. Equation of one Tangent to the Curve<br />
  93. 93. Image of Tangent to Curve<br />

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